Welcome to another Matholger video. When it comes to mathematics, most people think that all the really good and simple stuff has already been discovered a long, long time ago. Right?
Triangles, Pythagoras, all this stuff has been picked over for thousands of years. How can anything truly nice and easy have gone unnoticed until now? Well, not so!
:) Let me tell you about a wonderful little discovery about triangles that was only made 10 years ago. Here we go. Start with any triangle.
Mark the midpoints of the sides. And draw in the three medians of the triangle that end in these midpoints. These guys here.
And looks are not deceiving. The three medians always intersect in a point. And so the medians chop the triangle into six smaller triangles, usually all different in shape and size.
This fact has been known since time immemorial and mathematicians have been staring at this diagram for just as long. Now highlight two neighbouring triangles, those two. Insert a hinge right there at that midpoint.
Fold up. Nice. Let’s do the same with these two triangles here.
Actually, wait, before I fold things up a second time, can you already guess where all this is going? Yes? No?
Well, if we fold things up now we get EXACTLY the same triangle as before. Have a look. And now you can definitely guess what comes next, right?
Yes, put in a hinge, fold up and you get the same new triangle a third time. Tada, and that’s the new discovery. Very simple and very beautiful, isn’t it?
Well, there is a bit more. Let’s call the new triangle the folded triangle. The common side of the red and blue up there, that’s a median of the folded triangle.
Right? Pretty obvious. And, there is the second median.
And the third. Okay, now fold up the folded triangle. Great.
Now compare this new folded folded triangle to the original triangle. Notice anything special? Yes, the folded folded triangle is just a scaled down version of the original triangle.
Interesting and very beautiful isn’t it. And it really looks like this was only discovered for the first time 10 years ago. Amazing, right?
But of course as with most mathematics, a lot more beauty is hiding in the `why’. And so, let me also show you why all this works. Lot’s of nice AHA moments ahead and so make sure to stick around.
Actually we’ll prove more than just the medians meeting in the controid. We’ll also show that the centroid splits every median into a long and a short part. and that the long blue part is always exactly double the length of the short green part.
There. The blue is exactly twice as long as the green. Again, that’s true for any median, the blue is always twice as long as the green part.
There blue is twice as long as the green. And again, blue is twice as long as the green. Here is a nice way to see why all this is the case.
Scaling our triangle down by a factor of 3 gives this little triangle. And, you’ve probably all seen this before, we can now tile the large triangle with copies of the little triangle like this. Okay.
Here is one of the medians of the little grey triangle at the bottom. From the tiling it’s clear that if we extend this median up we end up with the median of another little triangle. That one here.
There extend. And extending further we get the median of the little triangle at the very top. And now of course, together the medians of the three little triangles form the median of the large triangle AND this median passes through the special point right in the middle of the tiling, AND the blue part of the median is twice as long as the green part.
AND, of course, the same is true for all three medians. Easy :) Okay, so medians all meet in the point right at the center of the tiling and are all cut by this point in the ratio of 1:2. Proof complete.
Super nifty, right? :) Actually with these two properties under out belt, it’s very easy to now also prove that the three ways of folding give the same folded triangle and that, furthermore, the folded folded triangle is a scaled down version of the original triangle. Here we go.
If these three parts really all fold up into the same triangle, then obviously all three parts, red, green and blue have to have the same area, exactly one third of the area of the original triangle. Let’s first see why that is true. Easy, The area formula for triangles is 1/2 base times height.
Well then, the blue triangle and the original triangle have the same base. And because of the 1:2 ratio in which the medians are cut by the centroid, it’s clear that the blue triangle’s height is 1/3 that of the original triangle. And, therefore, also the area of the blue triangle has to be exactly 1/3 of the area of the original triangle.
And of course the same is true for the red triangle, and the green triangle. Nice. Have you ever done any origami before?
For example, have you ever folded a paper crane? Well when you unfold the crane you get this interesting crease pattern which really is the blueprint for the crane design. Similarly, our median intersection pattern is a blueprint for everything that happens in our triangle folding adventure.
Everything that we come across in terms of little triangles, distances and angles is present in this blueprint. Let’s have a closer look. Let’s first highlight the angles around the centroid.
These two opposite angles are the same. So are these. and these and so on.
Okay, now any other angle visible in this diagram is composed of the angles in the middle. For example, let’s take one of the angles at the bottom. Put in a parallel to the bottom line through the centroid And so the angle we are interested in is equal to the orange angle plus the green angle.
Another example, this angle here. After drawing this parallel here it’s clear that our angle is equal to the grey angle plus the blue angle. Doing this angle chasing for the whole diagram gives this.
Pretty spectacular, hmm? Reminds me a little bit of an Australian aboriginal dot painting. Okay, get ready for an AHA moment.
Let’s do the folding. Okay, these are the three folded up triangles. They all look the same.
But to be sure that they really are, let’s check the angles at the vertices. There all the same and the same. And since we already checked that all three triangles also have the same area, we can be sure that all three triangles are congruent, that all three triangles are the same.
Okay, now let’s fold a second time We already know that these triangles are all the same. But we still need to check that this folded folded triangle is similar to the original triangle. For this we have to check that corresponding angles are the same.
Beauuuutiiiiiful :) Proof complete :) Now, let me just show you the folding action again in a slightly different way. As you’ll see, in a way, folding a triangle is really just turning it inside out, with a bit of scaling down thrown into the mix. And then folding again turns things inside out again and returns us to the original triangle.
We are back to where we started from. So, folding is really just a nifty way of turning a triangle inside out. Here are a couple more interesting observations.
First, isosceles triangles always fold into isosceles triangles. and an equilateral triangle folds into another equilateral triangle. What about right triangles.
Well let’s see. Folding a right triangles does not give a right triangle. However, since folding turns triangles inside out we can still find the right angles in the middle of the folded triangle.
There are the right angles. I can see some of you serious triangle lovers bouncing up and down in your seats :) Being well versed in the ways of the triangle, you will have immediately spotted another way of seeing pretty much at a glance why the three folded triangles are the same. Here is that second proof that quite a fwe of you will have come up with on your own.
This one just takes a couple of seconds. Color in the long median parts like this. Let’s fold So two of the sides of this folded triangle coincide with two of the coloured parts yellow and orange.
What can we say about the remaining black side at the bottom? let’s unfold again. Well, we know that this green segment is half as long as the blue.
Right, remember that? :) But then the remaining side of the folded triangle consists of two copies of the green and so the remaining side is as long as the blue side. nice.
And so the different sides of this folded triangle are just the three coloured long median parts of the original triangles. Since this is the same for all three folded triangles, all have to be the same. How nice is that :) Finally, here is the article by Lee Sallows about his discovery in the December 2014 issue of the Mathematics Magazine.
Just one page :) And so it really is still possible even today after thousands of years of triangle spotting to discover new simple and beautiful mathematics in this area. And that’s all for today’s mini Mathologer. Flat out with work at uni work at the moment and so didn’t have time for a Kurosawa length Mathologer video.
Still a very nice topic I think and I hope you enjoyed it :) Oh, and by the way, which of the two proofs do you prefer and why? The dot painting leisurely one which really takes in a lot the sights off the beaten track? Or the one-glance one at the end that goes straight for the kill?
